∫ 0 π cos ϕ d ϕ r 2 + c 2 − 2 c r sin θ cos ϕ {\displaystyle \int _{0}^{\pi }{\frac {\cos \phi \,d\phi }{\sqrt {r^{2}+c^{2}-2cr\sin \theta \,\cos \phi }}}}
= 2 R 1 ∫ 0 π 2 ( 2 cos 2 ϕ − 1 ) d ϕ 1 − R 1 2 − R 2 R 1 2 cos 2 ϕ {\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\int _{0}^{\frac {\pi }{2}}{\frac {\left(2\cos ^{2}\phi -1\right)\,d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\cos ^{2}\phi }}}}
= 2 R 1 ∫ 0 π 2 2 ( cos 2 ϕ − R 1 2 R 1 2 − R 2 ) − 1 + 2 R 1 2 R 1 2 − R 2 1 − R 1 2 − R 2 R 1 2 cos 2 ϕ d ϕ , {\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\int _{0}^{\frac {\pi }{2}}{\frac {2\left(\cos ^{2}\phi -{\frac {\mathrm {R} _{1}^{2}}{\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}}\right)-1+{\frac {2\mathrm {R} _{1}^{2}}{\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}}}{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\cos ^{2}\phi }}}\,d\phi ,}
= 2 R 1 { 2 1 − R 2 R 1 2 [ ∫ 0 π 2 d ϕ 1 − R 1 2 − R 2 R 1 2 sin 2 ϕ − ∫ 0 π 2 1 − R 1 2 − R 2 R 1 2 sin 2 ϕ d ϕ ] − ∫ 0 π 2 d ϕ 1 − R 1 2 − R 2 R 1 2 sin 2 ϕ } {\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\left\{{\frac {2}{1-{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}}}\left[\int _{0}^{\frac {\pi }{2}}{\frac {d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}}-\int _{0}^{\frac {\pi }{2}}{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}\,d\phi \right]-\int _{0}^{\frac {\pi }{2}}{\frac {d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}}\right\}}
= 4 R 1 ( 1 − R 2 R 1 2 ) [ { log 4 R 1 R + 1 2 2 2 R 2 R 1 2 ( log 4 R 1 R − 2 1.2 ) + . . . } − { 1 + 1 2 R 2 R 1 2 ( log 4 R 1 R − 1 1.2 ) + 1 2 ⋅ 3 2 2 ⋅ 4 R 4 R 1 4 ( log 4 R 1 R − 2 1.2 − 2 3.4 ) + . . . } ] {\displaystyle ={\frac {4}{\mathrm {R} _{1}\left(1-{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\right)}}\left[\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}+{\frac {1^{2}}{2^{2}}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}\right)+\,.\,.\,.\,\right\}-\left\{1+{\frac {1}{2}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {1}{1.2}}\right)+{\frac {1^{2}\cdot 3}{2^{2}\cdot 4}}{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}-{\frac {2}{3.4}}\right)+\,.\,.\,.\,\right\}\right]} − 2 R ′ { log 4 R 1 R + 1 2 2 2 R 2 R 1 2 ( log 4 R 1 R − 2 1.2 ) + 1 2 ⋅ 3 2 2 2 ⋅ 4 2 R 4 R 1 4 ( log 4 R 1 R − 2 1.2 − 2 3.4 ) + . . . } {\displaystyle -{\frac {2}{\mathrm {R} ^{\prime }}}\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}+{\frac {1^{2}}{2^{2}}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}\right)+{\frac {1^{2}\cdot 3^{2}}{2^{2}\cdot 4^{2}}}{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}-{\frac {2}{3.4}}\right)+\,.\,.\,.\,\right\}}
[Cayley, 'Ell. Func.,' p. 54.]
= 2 R 1 { log 4 R 1 R − 2 + R 2 R 1 2 5 log 4 R 1 R − 7 4 + R 4 R 1 4 162 log 4 R 1 R − 225 128 + e t c . } . {\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-2+{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}{\frac {5\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-7}{4}}+{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}{\frac {162\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-225}{128}}+etc.\right\}.}
Now
R 1 2 = 4 c 2 − 4 c R cos χ + R 2 {\displaystyle \mathrm {R} _{1}^{2}=4c^{2}-4c\mathrm {R} \cos \chi +\mathrm {R} ^{2}}
= 4 c 2 ( 1 − R c cos χ + R 2 4 c 2 ) {\displaystyle =4c^{2}\left(1-{\frac {\mathrm {R} }{c}}\cos \chi +{\frac {\mathrm {R} ^{2}}{4c^{2}}}\right)}
= 4 c 2 ( 1 − s cos χ + s 2 4 ) {\displaystyle =4c^{2}\left(1-s\cos \chi +{\frac {s^{2}}{4}}\right)}
Therefore
log 4 R 1 R = log 8 s − s 2 cos χ − s 2 4 cos 2 χ 2 − s 3 8 cos 3 χ 3 − {\displaystyle \log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}=\log {\frac {8}{s}}-{\frac {s}{2}}\cos \chi -{\frac {s^{2}}{4}}{\frac {\cos 2\chi }{2}}-{\frac {s^{3}}{8}}{\frac {\cos 3\chi }{3}}-\,} &c.
= l + 2 − s 2 cos χ − s 2 8 cos 2 χ − s 3 24 cos 3 χ − {\displaystyle =l+2-{\frac {s}{2}}\cos \chi -{\frac {s^{2}}{8}}\cos 2\chi -{\frac {s^{3}}{24}}\cos 3\chi -\,} &c.
R 2 4 R 1 2 = s 2 16 + s 3 16 cos χ + s 4 64 ( 1 + 2 cos 2 χ ) + {\displaystyle {\frac {\mathrm {R} ^{2}}{4\mathrm {R} _{1}^{2}}}={\frac {s^{2}}{16}}+{\frac {s^{3}}{16}}\cos \chi +{\frac {s^{4}}{64}}\left(1+2\cos 2\chi \right)+\,} &c.
![{\displaystyle ={\frac {2}{\mathrm {R} _{1}}}\left\{{\frac {2}{1-{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}}}\left[\int _{0}^{\frac {\pi }{2}}{\frac {d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}}-\int _{0}^{\frac {\pi }{2}}{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}\,d\phi \right]-\int _{0}^{\frac {\pi }{2}}{\frac {d\phi }{\sqrt {1-{\frac {\mathrm {R} _{1}^{2}-\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\sin ^{2}\phi }}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87e73846bcdc5ebf86e8ac7340a17beccccbd77e)
![{\displaystyle ={\frac {4}{\mathrm {R} _{1}\left(1-{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\right)}}\left[\left\{\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}+{\frac {1^{2}}{2^{2}}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}\right)+\,.\,.\,.\,\right\}-\left\{1+{\frac {1}{2}}{\frac {\mathrm {R} ^{2}}{\mathrm {R} _{1}^{2}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {1}{1.2}}\right)+{\frac {1^{2}\cdot 3}{2^{2}\cdot 4}}{\frac {\mathrm {R} ^{4}}{\mathrm {R} _{1}^{4}}}\left(\log {\frac {4\mathrm {R} _{1}}{\mathrm {R} }}-{\frac {2}{1.2}}-{\frac {2}{3.4}}\right)+\,.\,.\,.\,\right\}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb60af078a7dfcb92694a0acbb34341c5c0c00f6)
&c.
&c.
&c.
